261 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

Improved Rotated Finite Difference Method for Solving 

Fractional Elliptic Partial Differential Equations 

Abdulkafi Mohammed Saeed* 

Department of Mathematics , College of Science, Qassim University, Saudi Arabia 

Email: abdulkafi.ahmed@qu.edu.sa, Email: abdelkafe@yahoo.com 

 

 

Abstract 

Real life problems with fractional partial differential equations (FPDE's) are of great importance, since 

fractional differential equations accumulate the whole information of the function in a weighted form. This has 

many applications in physics, chemistry, engineering, etc. For that reason, we need a method for solving such 

equations, effectively, easy use and applied for different problems. The objective of this paper is to solve 

fractional elliptic partial differential equations, by using new accelerated version of rotated five point’s 

approximation method. Experiment results of the test problem are given in order to confirm the superiority of 

our proposed method. 

Keywords: Rotated Finite Difference Approximation Method; Fractional Elliptic Partial Differential Equations.  

1. Introduction  

Fast computational methods for solving partial differential equations using finite difference schemes derived 

from skewed (rotated) difference operators have been extensively investigated over the years. These Iterative 

methods based on the rotated finite difference approximations have been shown to be much faster than the 

methods based on the standard five-point formula in solving the partial differential equations which is due to the 

formers’ overall lower computational complexities ([1,2,3,4,5]). Fractional Partial Differential Equations 

(FPDE's) can be seen as a generalization of the classical partial differential equations (PDE's) in the sense that it 

takes into account the memory and hereditary properties of the physical phenomena ([6,7]). As it was in the 

classical PDE's there is no general method that can be used in solving FPDE's. Numerical solution of FPDE's 

has received great progress in the recent years ([8,9]).  

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

262 
 

The time and space-fractional partial differential equation describe transport dynamics in complex systems 

governed by anomalous dispersion and non-exponential relaxation [10].  

Because of complexity in the theoretic analysis of numerical approximation of fractional systems, the common 

approach is to apply the finite difference method to discretize fractional derivative operators, and then obtain the 

numerical solutions of the fractional partial differential equations. Furthermore, Goloviznin and his colleagues 

[11] developed a numerical method for solving some 1-D equations with fractional derivatives. 

The paper is organized in five sections: Section 2 describes the formulation of the Rotated Point Iterative 

Method for solving the fractional Poisson’s equation.  In Section 3, the proposed accelerated version of rotated 

five point’s approximation method will be given. In Section 4, the numerical results are presented in order to 

show the efficiency of the new proposed method. Finally, the conclusion is given in Section 5. 

2. Formulation of the Rotated Point Iterative Method for solving the Fractional Poisson’s equation   

Consider the Poisson’s equation in the form: 

( ) ( , ) ( , ),
α α

α α

∂ ∂
− + =
∂ ∂

u x y F x y
x y       

( , )∈x y D                                      (2.1) 

Where:  {( , ) : ( , ) [0, ] [0, ]}= ∈ ×D x y x y L L .  

Subject to the Dirichlet-boundary conditions: 

(0, ) ( ,0) ( , ) 0,u y u x u L y= = = and ( , ) ( )u x L g x=                                         

Beibalaev and his colleagues [8] considered the fractional Poisson’s equation in the form: 

( ) ( , ) ( , )u x y F x y
x y

α α

α α

∂ ∂
− + =
∂ ∂

, where 1 2,α≤ ≤                                                                   (2.2) 

subject to the same Dirichlet-boundary conditions of equation (2.1). Now, we consider the corresponding 

fractional order Elliptic by the form:  

( ) ( , ) ( , )∂ ∂
− + =
∂ ∂

u x y F x y
x y

α β

α β
, where  (1 , 2)α β≤ ≤                                                         (2.3) 

It can be seen that equation (2.3) is a generalization to equations (2.1) and (2.2). 

The simplest standard five-point finite difference approximation of the Laplacian is  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

263 
 

ij
jiijjijiijji fhO

y
u

x
uh

h
uuu

h
uuu

=−







∂
∂

+
∂
∂

−
+−

+
+− −+−+ )(

12
122 4

4

4

4

4
2

2
1,1,

2
,1,1                          (2.4) 

Here, ),( jiij yxuu = .  Another approximation to equation (2.1) can be derived from the rotated five-point finite 

difference approximation to give [1] 

1, 1 1, 1 1, 1 1, 1
2

4 4 4
2 4

2 2 4 4

4
2
1 1                   ( ) .
2 12

i j i j i j i j ij

ij

u u u u u
h

u u uh O h f
x y x y

+ + − − + − − ++ + + −

  ∂ ∂ ∂
− + + − =  

∂ ∂ ∂ ∂   

                                                     (2.5) 

In order to obtain the finite difference approximation of the fractional order equation (2.2), we use the treatment 

introduced in [8] for approximate Caputo’s fractional derivative of order ,α (1 2)α≤ ≤  in the form: 

1, 1,2
( , ) ,

(3 )
α

αα
+

+ −− +
≈

−i

i j ij i jc
i jx

u u u
D u x y

h
                                         (2.6a) 

, 1 , 12
( , )

(3 )
α

αα
+

+ −− +
≈

−j

i j ij i jc
i jy

u u u
D u x y

h
                                          (2.6b) 

We can observe that for equation (2.1), the corresponding finite difference approximation of the Caputo’s 

fractional order derivative of order α is:  

4 (4)
1, 1, max ( )2

( , )
12(3 )(3 )

α
α

α αα
+

−
+ −− +

≈ +
−− i

ii j ij i j c
i jx

h u xu u u
D u x y

h
 

(2.7a) 

and 

4 (4)
, 1 , 1 max ( )2

( , )
12(3 )(3 )

α
α

α αα
+

−
+ −− +

≈ +
−− j

ii j ij i j c
i jy

h u xu u u
D u x y

h
    (2.7b) 

Therefore, the standard five-point finite difference approximation of equation (2.2) can be written as: 

1, 1, , 1 , 12 2
(3 ) (3 )

+ − + −− + − +
+ = −

− −
i j ij i j i j ij i j

ij
u u u u u u

f
h hα αα α

 

which can be rearranged as in the form 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

264 
 

1, 1, , 1 , 14 (3 ) αα+ − + −− − − − = −ij i j i j i j i j iju u u u u h f
                                

(2.8) 

By the same manner, the rotated five-point finite difference approximation can be written as: 

1, 1 1, 1 1, 1 1, 14 (3 ) αα+ + − − − + + −− − − − = −ij i j i j i j i j iju u u u u h f
                       

(2.9) 

There are two ways to approximating equation (2.3), the first one by using Caputo’s formula which is replaced 

by a finite sum of integrals at the discretization points, and approximate the second order derivative by using the 

standard five-point finite difference formula (2.8) or rotated five-point finite difference formula (2.9). If the 

standard five-point finite difference is used, then the finite difference formula of Caputo’s fractional derivative 

will take the form: 

1
1 2

2 1
0 2

0

1 ( , )( , ) ( )
(2 )

+
−

− −

=

∂
≅ −
Γ − ∂∑∫

k

k

i xc
x x

k

u s yD u x y x s ds
s

α α

α
 

1

0 , 1, , 1,
0

1 ( 2 )
(3 )

−

− + − − −
=

≅ − +
Γ − ∑

i
c

x i j k i k j i k j i k j
k

D u b u u u
h

α α
α α                      (2.10) 

where 
2 2[( 1) ]− −= + −kb k kα α α

,   let 
*

(3 )
−

=
Γ −

s
s

bb
h

β

β β    ,  (3 )
−

=
Γ −

k
k

bb
h

α

α α   

Then, the finite difference scheme for equation (2.3) will be given in the form: 

11
*

1, , 1, , 1 , , 1
0 0

( 2 ) ( 2 )
−−

− + − − − − + − − −
= =

− + + − + =∑ ∑
ji

k i k j i k j i k j s i j s i j s i j s ij
k s

b u u u b u u u f
 

where ( , ).=ij i jf f x y  If ,=α β  we can see that:  * 1 .
(3 )
−

= =
Γ −s kb b b

h
α

α α  

The second way to approximating equation (2.3) by using Grunwald-Letnikov (G-L) approximation [12] as the 

following: 

0 ,0
0

1( , ) lim ,
( )

−
−∆ →

=

=
∆ ∑

N
R L

x i j k N k jx
k

D u x y w u
x

α α

        

( )
( 1) ( )
Γ −

=
Γ + Γ −k

kw
k

α α
α  

0
0 0

0

( , )
( , ) ( , ) ,

( 1 )

−
− =

=

∂
= −

Γ + − ∂∑
kN k

c R L x
x x k

k

u x yxD u x y D u x y
k x

α
α α

α  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

265 
 

1

0 1,
0

1( , ) ( ).
+

−
− +

=

= +∑
i

G L
x i j k i k j

k

D u x y g u o h
h

α α
α

 

By using the standard five-point finite difference approximation (2.8), equation (2.3) can be written in the form: 

11

1, 1,
0 0

++

− + − +
= =

+ =∑ ∑
ji

k i k j s j s i ij
k s

g u g u fα β

                                      (2.11) 

where     1 ;= −k kg r zα α
 2 ;= −s kg r zβ β

1
1 ;=r

hα 2
1 ;= kr z

h
β

β  0 1;=zγ 1 ;= −zγ γ
 

( 1)...( 1) ; 1
!

− − +
= >k

kz k
k

γ γ γ γ

 

Furthermore, by using the rotated five-point finite difference approximation (2.9), equation (2.3) can be written 

in the form: 

11

1, 1 1, 1
0 0

++

− + + − + +
= =

+ =∑ ∑
ji

k i k j s j s i ij
k s

g u g u fα β

                                   (2.12) 

where     1 ;= −k kg r zα α
 2 ;= −s kg r zβ β

1
1 ;=r

hα 2
1 ;= kr z

h
β

β  0 1;=zγ 1 ;= −zγ γ
 

( 1)...( 1) ; 1
!

− − +
= >k

kz k
k

γ γ γ γ
 

3. The proposed accelerated version of rotated five point’s approximation  method   

It’s well known that in the finite difference treatment the PDE's or the FPDE's are replaced by an algebraic 

system of equations which can be written as the form 

,=Au f                                                                                     (3.1) 

where, A is 2 2(N 1) (N 1)− × − coefficients matrix, u  and f  are two 2(N 1) 1− ×  matrices, where

,1 ,2 , 1[ , ,..., ]−= T
j j j Nu u u u and ,1 ,2 , 1[ , ,..., ] ,−= T

j j j Nf f f f  1, 2,..., 1.= −j N  

It is well known that the computational molecule of standard finite difference approximation for the classical 

(integer) case of PDE's can be represented as in figure 1 whereas the computational molecule of standard finite 

difference approximation for FPDE's can be represented as in figure 2. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

266 
 

 

Figure 1: Computational molecule of Eq. (2.4)                 Figure 2: Computational molecule of Eq. (2.8) 

 

Also, we can observe that for the rotated five-point finite difference approximation the following 

transformations take place 

1 1 1i , j i , j± → ± ±  

1 1 1i , j i , j± → ±   

2h h.→  

Therefore, the computational molecule of the rotated five-point finite difference approximation for PDE and 

FPDE can be shown in figure 3 and figure 4 respectively. 

 

 

 

      

Figure 3: Computational molecule of Eq. (2.5)                         Figure 4: Computational molecule of Eq. (2.9) 

It is clear that the coefficients matrix for the fractional order case 0f equations (2.8) and (2.9) have the same 

structure as in equations (2.4) and (2.5) except the free column f  in the right side of system (3.1). 

Theoretically, it can be seen that the coefficients matrices resulting from systems (2.4) and (2.5) are non-

singulars, so the system (3.1) has unique solution ([5]). In addition to that the coefficients matrix A  is strictly 

diagonal dominant, then A is non-singular and systems (2.8) and (2.9) have a unique solution for (1 < α, β < 2). 

Since it is well known that preconditioners play a vital role in accelerating the convergence rates of iterative 

methods, several preconditioned strategies have been used for improving the convergence rate of the iterative 

methods derived from the standard and skewed (rotated) finite difference operators ([4], [5]). A well-designed 

i+1,j i-1,j 

i,j-1 

i,j 

i,j+1 

i+1,j 0,j 

i,0 

i,j 

i,j+1 

i+1,j+1 

i-1,j-1 
i+1,j-1 

i,j 

i-1, j+1 

i+1,j+1 

i-1,0 
i+1,j-1 

i,j 

0, j+1 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

267 
 

preconditioning of the PDE and FPDE problems reduces the number of iterations to reach convergence. 

Dramatic improvements are possible, but the difficulty is to construct the suitable preconditioner.  In general, a 

good preconditioner should satisfy the following prosperities: the first one is that, the preconditioned system 

should be easy to solve and the second one is that the preconditioner should be cheap to construct and apply. 

Usually the system (3.1) is large and the matrix A is sparse. Furthermore, matrix A  can be write as 

A D L U= − −                                                                                (3.2) 

where D is diagonal matrix  A, L− is strictly lower triangular parts of  A and U− is 

strictly upper triangular parts of A. A preconditioner ( )+I ML  where 0 2≤ <M  is used to modify the 

original system (3.1) to the following system: 

( ) ( )+ = +I ML Au I ML f                                                           (3.3) 

A preconditioner ( )= +P I ML is a matrix that transforms the original system (3.1) into new system (3.3) that 

is equivalent in the sense that it has the same solution, but that has more favourable spectral properties. 

4. Numerical Results and Discussion 

In the first part of this section, we have compared between the spectral radiuses of the iteration matrix 

corresponding to the resulting system of rotated five-point finite difference (original system) and the 

preconditioned system for different values of  ,α β such that (1 , 2)α β≤ ≤ . Table 1 shows the comparison of 

the spectral radius between the original and the preconditioned systems. Clearly it can be seen that the spectral 

radius of the preconditioned system is smaller compared to the original system, thus justifying the superiority of 

the preconditioned system against the original system which is quite agreeing with the results obtained in the 

previous work ([3,4,8]).  

Table 1: Comparison of the spectral radiuses of the iteration matrix corresponding to the original and the 

preconditioned systems 

 

 

 

 

 

N 
order Original system Preconditioned system 

α  β  ( )Aρ  ( )PAρ  

42 1.2 1.4 0.053 0.041 

74 1.4 1.4 0.091 0.083 

114 1.6 1.4 0.141 0.112 

186 1.8 1.4 0.184 0.171 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

268 
 

The second part of this section has discussed the numerical solution of the following modal problem using the 

proposed iterative method: 

2 2( , ) ( , ) 10[4 ( 2 )]sin+ ++ = − + −c c
x y

D u x y D u x y x x yβ β π π
 

defined in {0 0.5, 0 1}= < < < <D x y  with the boundary condition 0Γ =u
 
such that: Γ is the boundary of 

the area D, and the fractional order is 1 2 .< ≤β   

Table 2: Solution of the modal problem for 1.5=β by the original and the preconditioned  systems 

 

 

Numerical data of the original and the preconditioned systems are summarized in tables (2-3) for two fractional 

orders for 1.5=β and  2 .=β  It can be observed that in all cases illustrated the increase in the fractional order

2=β reduces the magnitudes of the peaks.  

Throughout the two sections of our experiments, the results reveal that the proposed preconditioned is superior 

to the original system in solving fractional elliptic partial differential equations.  

 

 

 Original system  Preconditioned system 

 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 

0 0 0 0 0 0 0 0 0 0 0 0 0 

0.1 0 1.846 2.831 3.046 1.984 0 0 1.561 2.503 2.883 1.552 0 

0.2 0 2.692 5.411 5.662 3.784 0 0 2.341 4.854 5.404 3.334 0 

0.3 0 3.147 7.604 7.934 4.527 0 0 2.733 6.931 7.663 4.416 0 

0.4 0 4.533 7.599 7.962 5.681 0 0 3.883 7.201 7.533 5.334 0 

0.5 0 5.703 8.104 8.544 6.571 0 0 4.914 7.995 8.212 6.204 0 

0.6 0 6.342 8.212 8.704 7.364 0 0 5.683 8.113 8.673 6.677 0 

0.7 0 5.425 6.425 6.225 6.225 0 0 4.842 5.524 5.791 5.875 0 

0.8 0 3.843 4.511 4.341 4.542 0 0 2.661 3.641 3.973 4.341 0 

0.9 0 1.846 2.831 3.046 1.984 0 0 1.543 2.492 2.785 1.473 0 

1 0 0 0 0 0 0 0 0 0 0 0 0 

x y 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

269 
 

Table (3): Solution of the modal problem for 2=β by the original and the preconditioned systems 

 

 

5. Conclusion and Future Works 

In this paper, we have formulated new preconditioned iterative method based on rotated finite difference method 

for solving fractional elliptic partial differential equations. From observation of all experimental results, it can 

be concluded that the proposed scheme may be a good alternative to solve fractional elliptic partial differential 

equation and many other numerical problems.  The idea of this proposed method can be extended to group 

iterative solver which will be reported separately in the future. 

Acknowledgements 

 Financial support provided by Qassim University for the completion of this research is gratefully 

acknowledged. 

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[1] A. R. Abdullah, “The Four Point Explicit Decoupled Group (EDG) Method: A Fast Poisson Solver” 

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[2]  A. Gunawardena, S. Jain, L. Snyder, ”Modified iterative methods for consistent linear systems”. 

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 Original system  Preconditioned system 

 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 

0 0 0 0 0 0 0 0 0 0 0 0 0 

0.1 0 0.246 0.387 0.365 0.234 0 0 0.203 0.324 0.324 0.179 0 

0.2 0 0.383 0.522 0.534 0.461 0 0 0.334 0.462 0.475 0.403 0 

0.3 0 0.527 0.924 0.964 0.625 0 0 0.503 0.856 0.922 0.556 0 

0.4 0 0.533 1.024 1.013 0.691 0 0 0.481 0.987 0.989 0.603 0 

0.5 0 0.564 1.171 1.142 0.717 0 0 0.524 1.043 1.031 0.601 0 

0.6 0 0.531 1.141 1.122 0.677 0 0 0.499 1.044 0.987 0.662 0 

0.7 0 0.424 0.993 0.976 0.504 0 0 0.375 0.985 0.921 0.478 0 

0.8 0 0.363 0.651 0.691 0.463 0 0 0.322 0.622 0.584 0.357 0 

0.9 0 0.231 0.373 0.362 0.234 0 0 0.203 0.323 0.304 0.178 0 

1 0 0 0 0 0 0 0 0 0 0 0 0 

x y 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  1, pp 261-270 

 

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